Eternal 1-security number of the join, and composition of some graphs
Michael P. Baldado, Sergio R. Canoy · Applied Mathematical Sciences · 2014
Let G be a graph. An eternal 1-secure set in a graph G is a set S0 ⊆ V (G) with the property that for any k ∈ N and any sequence � v1 ,v 2 ,... , vkof vertices of G, there exists a sequenceu1 ,u 2 ,... , uk� of vertices of G with ui ∈ Si−1 and either ui equal to or adjacent to vi, such that each set Si =( Si−1\{ui}) ∪{ vi} is dominating in G. The eternal 1-security number of G, denoted by σ1(G), is the minimum cardinality of an eternal 1-secure set in G. In this paper, the eternal 1-security numbers of the join and compo- sition of some graphs are given.